Some Features of the Solutions of Ku aka and Vargaprak ti

16 Pages·2017·0.79 MB·English

Indian Journal of History of Science, 52.1 (2017) 1-16 DOI:10.16943/ijhs/2017/v52i1/41294
Some Features of the Solutions of Kuaka and Vargaprakti
A K Bag*
(Received 15 March 2016)
Abstract
The expertise in Kuaka and Vargaprakti, the methods used for the solution of first and second
degree indeterminate equations respectively, were considered pre-requisite qualifications of an Acharya
in ancient and medieval India.. For solution of Kuka of the type; b y = a x + c, the values of were
approximated from the successive divisions of a by b as in HCF process and the number of steps was
reduced with choice of a desired quantity [mati] at any step, even or odd . The solution of Vargaprakti of
the type, N x 2± c = y 2 [where N = a non-square integer, and c = kepa quantity] was in the manipulation
of the value of √N→ based on two set of arbitrary values for x , y, and c and their cross multiplication
when c = ± 1, ± 2, ± 4, as given by Brahmagupta (c. 628 CE). The solution was concretized by Jayadeva
[1100 CE] and Bhāskara II [1150 CE] by a process, known as Cakravāla. The number of steps used in
Cakravāla is much lower than the regular and half-regular expansions for √N used by Euler and Lagrange.
The minimization property of Cakravāla is unique and the method may be treated as one of the major
achievements of Indian mathematics in the history of solution of second degree equations.
Key words: ryabhaa, Bhskara I, Bhskara II, Brahmagupta, Cakravla, Diophantus, Euler,
Half-regular expansion, Jayadeva, Kepa, Kuaka, Lagrange, Minimization properties, Nryaa, Pierre
de Fermat, Regular expansion, Vargaprakti
1. IntroductIon ekavara-bhāvitakaih/ācārya sa tantravidām
jňātaih varga-prakrtyā ca //
ryabhaa I, the pioneer siddhāntic
mathematician cum astronomer who was born
English translation:
in Kusumpura (near Patna) in 476 CE wrote his
ryabhaīya () at the age of twenty-three. He One who is well versed in [operations]
concretized his knowledge of arithmetic, algebra with the kuaka (pulverizer), kha (zero), a-
including pulverizer (kuaka) and geometry in dhana (negative and positive quantities), avyakta
his second chapter on mathematics (gaita). (unknown quantities), madhya-haraa (the
Brahmagupta, the first great mathematician elimination of the middle term), ekavara (one
of Indian history after ryabhaa I, wrote his unknown), bhāvita (equations involving products
Brāhmasphuasiddhānta (BSS) in 628 CE in of unknowns) and also varga-prakti (second
Ujjain at the age of thirty, and is the earliest degree equations) is [recognized as] a great teacher
known Indian mathematician to have separated (ācārya) among the specialists (tantravids).
algebra from mathematics (gaita). He described
The above verse shows that Brahmagupta
the qualifications of an ācārya (‘great teacher’) in
set a very high standard for qualifications of an
algebra, in the following words (BSS, xviii 2):1
ācarya in algebra. It was emphasized that he
kuaka-kha-adhana-avyakta-madhya haraa- should be expert in the operations of Kuaka and
*Editor, IJHS, Indian National Science Academy, Email: [email protected]
2 INdIAN JourNAL of HIStory of SCIENCE
Vargaprakti beside others. Both the operations śeaparaspara bhaktam matiguam agrāntare
had wide ramifications in both mathematics and kiptam /
astronomy.
adhaupari guitam antyayug ūnāgracchedabhājite
In this paper, I will discuss the features śeam /
of solutions of Kuaka of the type : b y = a x ± 1
adhikāgracchedaguamdvicchedaāgramadhikāgr
and by = ax ± c , and of Vargaprakti of the type:
ayutam // (Ā, Gaita, vs.32-33)
Nx2 ± 1 = y2 and Nx2±c= y2 as found in Indian
Tr. Divide the divisor (adhikāgra-
tradition.
bhāgahāra ) corresponding to the greater remainder
(adhikāgra), by the divisor (unāgra bhāgahāra)
2. KuaKa of the type: by = ax ± c
corresponding to the smaller remainder (unāgra);
The solution of indeterminate equations
the residue and the divisor corresponding to
of the type :
the smaller remainder being mutually divided
by = a x ± c, leads to: (śeaparaspara bhaktam); the residue (at any
stage) is to be multiplied by a desired integer
y = (a>b)….(1), or x = (b>a)..(2). (mati) to which the difference of the remainders
(kepa) is added (the number of partial quotients
The solution was actually manipulated by
being even) or subtracted (the number of partial
Āryabhaa I from the approximations of →
quotients being odd), the result when divided
in (1), and → in (2). by the penultimate remainder will give the final
quotient; the partial quotients, the mati and the
2.1. Āryabhaa I (b. 476 CE) final quotient are placed one below the other;
then, the mati is to be multiplied by the quotient
Āryabhaa I, the pioneer siddhāntik
above it to which the final quotient below it
mathematician, himself cited that he had his
is to be added (adhaupari guitam antyayug),
education in Kusumpura school (kusumpure
and the process (of multiplication and addition)
carcita jňānam, Ā, ii.1).The place has been
is continued; the last number obtained is then
identified in North India between Patna and
divided by the divisor corresponding to the smaller
Nalanda by Shukla (vide his edition of the text,
remainder; the residue is then multiplied by the
Āryabhaīya, Introduction p. xviii). Bhāskara I
divisor corresponding to the greater remainder to
referred to Āryabhaa I as an Āśmakīya, which
which the greater remainder is added; the result
indicates that he belonged to Aśmaka tribe or
will determine the number corresponding to the
country (MBh, Eng tr, p.2), and according to
two divisors.
commentator Nīlakantha he was born in that
country. the Aśmaka country has also been Explanation: Āryabhaa I might have
identified with Kerala by some scholars. been interested to find a number (N) , which
when divided by an integer (a) leaves a remainder
A. Rule: Āryabhaa I gives a rule in his Āryabhaīya
( r ), and by an integer (b) separately leaves a
for obtaining solution by mutual division of a 1
remainder (r ) .
and b as in HCF process (a, b are integers) and 2
is the knowledge of pulverization or kuakāra. Or, N = a x + r = b y + r ,
1 2
The rule (Āryabhaīya, Gaita, 32-33) runs
i.e., to solve: b y = a x ± (r - r ) accordingly
1 2
thus:
as r >r or otherwise,
1 2
adhikāgra bhāgahāram chindyāt unāgra
or b y = a x ± c , where c = (r - r ),.
bhāgahārea / 1 2
SoME fEAturES of tHE SoLutIoNS of KuaKa ANd VarGapraktI 3
(1) Solution of: by = a x + c other (here it is placed side by side result being
same), and the process of multiplication is to be
Āryabha a I proceeded with the
started from the mati (m) upwards multiplying
approximation → (a>b), where a and b were
with the upper quotient & the final quotient (q)
mutually divided as in HCF process, a and b being
as additive; the operation then is repeated and
integers. He kept c (i.e., r - r or r - r ) always
1 2 2 1 stopped after getting two final numbers. the
positive.
operation is same as in modern process.It may be
The rule says, when a and b are mutually represented as follows:
divided (a > b) , a being the dividend and b being
the divisor as in HCF Process of Division :
=

for t = 0, 1, 2, ..;
It gives → (penultimate convergent of
),which is undoubtedly an ingenious technique
for obtaining the equation : a x - b y = - c
1 1
(c = +ve or – ve depending on the even or odd
[where q ,q .…..q . are partial quotients, number of quotients).
1 2 n
and r , r , r ....r are corresponding remainders].
1 2 3 n Then, (x , y ) is the solution of :
1 1
b y = a x + c, where c = r -r .The desired number
If r = 0, then = q + …. = 1 2
n 1
N is found from:
(q ,q ,….,q ). Āryabhaa I however introduced
1 2 n
a unique method to find the approximation or N = a x + r = b y + r .
1 1 1 2
convergent of . In this method he could stop at
(2) Solution of : b y = a x – c, or a x = by + c
any point of the HCF process to compute a result
(b<a),
which is nothing but the penultimate convergent.
Then, → = (0, q , q , q , ) = (no.
What he did, he advised to multiply any remainder 1 2 3
of the division by a desired quantity (m), to which of quotients even or odd), or b y - a x = - c, or
1 1
the kepa quantity (c) is to be added or subtracted b y = a x - c giving a solution (x ,y ) for b y =
1 1 1 1
depending on the number of quotients even or a x - c.
odd respectively, and the result when divided
Āryabhaa I, however, did not specify
by the previous remainder gives a final quotient
the results for even or odd number of quotients,
(q). As a result, →(q , q , q , q , ). In short,
1 2 3 4 the details of which is of course clear from the
the quantities m and q were obtained from the commentary of Bhāskara I, which says, ‘add kepa
following, (c) when number of quotients are even, and subtract
when these are odd ; so is explained by schools
= q (n=no.of quotients as
(agrāntaram prakipya viśodhyam vā asya rāśeh
even or odd respectively),
śuddham bhāgam dāsyatīti / sameu kiptam
Then the rule says, the partial quotients: visameu śodhyam iti sampradāyāvicchedād
vyākhyāyate/) [Bh, ii.32-33 (bhāya of Bhāskara
(q , q , q q , ) are to be placed one below the
1 2 3 4
4 INdIAN JourNAL of HIStory of SCIENCE
I)]. Brahmagupta (BSS, xviii, 3-5, Eng. Tr. Datta
[quotients = 2, 3, 1, (number even);
& Singh,pt.2, pp.1-2) gave exactly the same
remainders (r ,r ,r )= 17, 9, 8 respectively];
method. 1 2 3
giving, = = 1 (m=mati = 1, final
B. Features of Āryabhaa I’s solution:
quotient= q =1, c = positive= + 1, number of
(i) for solution of : b y = a x + c, Āryabhaa I gave quotients being even);
an ingenious method actually manipulating
This leads to : 137 . 7 - 60. 16 = - 1, or 60 y =
1
( →) → (q ,q ,q ,q , ) = (a>b), 137 x + 1, giving x = 7, y = 16. This gives,
1 2 3 4 1 1 1
x = 7. y = 16, fixing the minimum solution
where m and q are found from: =q
of 60 y =137 x + 1.
(n=even or odd). The result is nothing but
the penultimate convergent of leading to the This suggests that is the penultimate
solution of : a x - b y = - c , or b y = a x + convergent of .
1 1 1 1
c (when number of quotients is even or odd,
c = kepa number). The value (x ,y ) gives (ii) → → 2 + = =
1 1
the solution of: b y = a x + c from which the
→ , [quotients
required number N is obtained.
(number odd): 2, 3, 1, 1. .; corresponding
(ii) For solution of : b y= a x - c, or a x= b y + c,the
remainders : 17, 9, 8, 1]; for =
original approximation ( →) → (b>a),
which is also the penultimate convergent = 8; then m = 9; q = final quotient = 8, the
leading to the solution of :b y - a x = - c, or
1 1 number of quotients being odd .
b y = a x - c (when number of quotients is
1 1
even or odd , c = kepa number). The values of This leads to : 137. 67 - 60. 153 = - 1, or 60.
(x , y ) gives the solution of b y = a x – c. Or (137 + 16) – 137.(60 + 7)= 1, or 60. 16 = 137.7
1 1 1 1
in other words, (x, y) gives the solution from + 1, or 60 y = 137 x + 1, giving x = 7, y =
1 1 1 1
which the required number N is obtained. 16.
(iii) Indicates that if (x , y ) is the solution of : then, x = 7, y = 16,fixes the solution of 60 y
1 1
b y = a x = ± 1 , then (cx , cy ) is the solution = 137 x +1.
1 1 1 1
of b y = a x ± c; and
1 1 The solution fixes the penultimate
(iv) Āryabhaa I managed to obtain the solution convergent as of (the number being even
of c (p q - q p ) = ± c, c being any kepa
n n-1 n n-1 or odd). this satisfies the relation (p q - q p ) =
quantity, for n = even or odd. n n-1 n n-1
±1 (for n = even or odd).The solution is same when
the number of quotients is even or odd..
C. Examples:
Now, N = 137 x + 8 = 137. 7 + 8 = 967, or
1. To find a number N such that N= 60 y + 7
N = 60 y + 7 = 60. 16 + 7 = 967.
=137 x + 8
This leads to : 60 y = 137 x + 1 (here r = 2. To solve : 60 y = 137 x – 1
1
7, r = 8, c = r -r = 8 – 7 = 1)
2 2 1 The equation reduces to : 137 x = 60 y + 1.
(i) → → 2 + = = → → = 0 + =
SoME fEAturES of tHE SoLutIoNS of KuaKa ANd VarGapraktI 5
of the rules are extremely interesting. He set
[quotients: 0, 2, 3, 1, 1, 1 (final quptient);
a large number of examples for the solutions
remainders : 60, 17, 9, 8, 1]; m is calculated from : of indeterminate equations for (1) and (2),
= = 1 (final quotient, no. of quotients keeping kepa quantity (c) as positive, following
Āryabhaa I. However, Bhāskara I emphasized
being even). This leads to : 60. 121 – 137. 53 = - 1,
more importance to the solution of : b y = a x - c,
or 60. 121 = 137. 53 – 1, or b y = a x – 1, or x =
53, y = 121 giving solution of 60 y = 137 x – 1. or y = straightway, because of its application
in solving astronomical problems, where a =
3. To find a number N such that N = 60 y = revolution number, b = civil days in a yuga, c =
137 x + 10 residue of the revolutions of planet. x = number
of days passed from the epochal point (ahargaa),
(a) → → 2 + =
and y = complete revolutions performed by the
planet.
→ →
[quotients (number odd): 2,3,1,1, ; B. Features of Bhāskara I’s solution:
corresponding remainders: 17, 9, 8,1; for (i) Dividend and the divisor (a and b) should
be prime to each other (hārabhājyau dau
= = 1; hence m = 18, q = 1].
syātām kuakāramtayorviduh / MBh, i. 41);
This leads to : 137. 10 - 60. 23 = - 10, or
(ii) For the solution of : b y = a x – c ( a<b); the
60. 23 = 137. 10 + 10;
mutual division of → ) leading to its
Comparing with 60 y = 137 x + 10, it gives, solution.
x = 10, y = 23 as the solution of 60 y = 137 x
Let : → = q + …….., (where
+ 10. Now ; N = 137 x + 10 = 137, 10 + 10 = 1
q =0). The first quotient being zero was
1380. 1
not effective in the calculation. Obviously,
(b) From Example C.I: x = 7, y = 16 is the solution
Bhāskara I concluded that the kepa number
of : 137 x + 1 = 60 y. For, c = 10, obviously,
is to be subtracted (apanīyam) for even number
x= (7. 10) = 70 = 10 (mod 60) For, 70 = 60. 1
of quotients,and added for odd number of
+ 10); y = (16. 10) = 160= 23(mod 137); For, quotients (MBh, i.42-44).
160 = 137. 1 + 23= 23 (mod 137);
(iii) Bhāskara I also suggests that if (x , y ) is
1 1
This shows that if (x =7, y = 16) is the solution the solution of : b y = a x – c, then (b - x ,
1
of : 137 x + 1 = 60 y, then (c x, c y) is the a - y ) is the solution of : b y = a x + c. Likewise,
1
solution of : 137 x + 10 = 60 y, for c = 10. if (x , y ) is the solution of : b y = a x + c, then
1 1
(b - x , a- y ) is the solution of : b y = a x - c.
1 1
2.2 Bhāskara I (c.600 CE)
(iv) He also explained that if (x , y ) is the solution
1 1
Bhāskara I imbibed his knowledge of of : a x - 1=b y , then (x = c x , y=c y ) is the
1 1 1 1
astronomy from his father, a follower of the school solution of : a x – c = b y (MBh, i.47);
of Āryabhaa I. He wrote his Mahābhāskarīya
(v) Bhāskara I also recommended that if x = x ,
(MBh), Āryabhaīya-bhāya (ĀBh) (in 629 CE) 1
y = y is the minimum solution of: b y = a x – c,
and Laghu-bhāskarīya (LBh) in order and used a 1
then the other solutions of the same equations
large number of problems relating to Kuaka.
are : x = b t + x , y = a t + y , for t = 1, 2, 3,
1 1
A. Bhāskara I’s clarification and modification .. (MBh. i.50)
6 INdIAN JourNAL of HIStory of SCIENCE
(vi) Bhāskara I had also the knowledge of successive 347 = 3091, a - y = 1397 -141= 1256, will be
1
convergents. the solution of 3438 y = 1397 x + 1.
Let = q + …….. 2. To solve ; 3438 y = 1397 x - 1
1
Then, Here, → = 0 +
=0, (convergents).
…, where ,..are the 1st, 2nd, 3rd
This shows that the Indian method always
convergent (or approximation) values of the calculated the value of penultimate convergent
of ., which gives: x = 347, y = 141 (n
rational number .
= even).
Bhāskara I’s application justifies his knowledge of
convergents. In his formula for declination,he 3. The residue of the revolutions of Saturn
uses two variants of the same result as being 24, find the ahargaa and the
under: revolutions made by Saturn [LBh,viii.17;
see also Shukla, MBh edition, p.30)
(a) r Sine δ = (MBh. iii. 6-7 ), and
Saturn’s revolution number= 146564, number
(b) R Sine δ = (MBh. iv. 25),
of civil days = 1577917599, both numbers
where δ =declination, λ= longitude. has an HCF = 4; dividing by 4, the number
of Saturn’s revolutions , and the civil days
The result , the fifth comvergent of
in a yuga are: 36641, 394479375; to find
(vide Example C.2 below) is used in the
the ahargana (x) and the Saturn’s revolution
formula for declination, it is quite likely that number (y); this leads to; y =
Bhāskara I had the knowledge of successive
Now, → = 0 +
convergents.
C. Examples: = =
1. To solve ; 3438 y = 1397 x - 1 = ; quotients:
According to Bhāskara I’s procedure, → 0, 10766, 15, 2, 7, 22, 2, 1 (final quotient),
remainders : 36641, 2369, 1106, 157, 7, 3, 1;
= 0 + = , where
m (mati is obtained from : =
partial quotients=0,2, 2, 5, 1, and 1 (final
quotient); the partial remainders = 1397, 644, = 1 for m = 27. This gives, 394479375.
109, 99, 10; m (mati) and the final quotient 1
32292 – 36641. 346688814 = - 24; This
is obtained from, (c becomes negative no. of
gives, 394479375 y = 36641. x - 24 where
quotients being even) : = 1 1 1
x = ahargaa=346688814, y = Saturn’s
1 1
(final quotient) for m = 10. This leads to: 1397. revolution=32292.
347 - 3438. 141 = 1, or 3438. 141 – 1397 .
347 = - 1, or 3438 y = 1397 x -1,which gives 2.3. Brahmagupta
1 1
x = 347, y = 141 as the required solution.
1 1 The most prominent of Hindu
This also indicates that is the penultimate
mathematicians belonging to school of Ujjain
convergent of .
was Brahmagupta. His Brāhmasphuasiddhānta
Bhāskara I suggests that then, b - x = 3438 – (BSS) was composed in 628 CE.
1
SoME fEAturES of tHE SoLutIoNS of KuaKa ANd VarGapraktI 7
A. Features of Brahmagupta’s solution: A Bhāskara II’s rules are far more simplified and
may be summarized thus. This in short:
(i) Recommended the same rule, as was prescribed
by Āryabhaa I for the solution of: b y = a x (i) for solution of b y = a x + c, Bhāskara II said
+ c (BSS, xviii.3-5); that the mutual division may be continued to
finish, i.e., till the last remainder is 1; then
(ii) For the solution of : b y = a x – c (a<b),
the sequence of quotients should follow with
Brahmagupta supported the method of
Āryabhaa I and Bhāskara I ,when first c and 0.e.g., → = (q , q , q , q , c/0)
1 2 3 4
quotient is zero and not effectively taken part
= q + where c = any
1
in the calculation. Brahmagupta categorically
number,leading to the solution of ; c (a x - by )
said, ‘Such cases become negative and positive 1 1
= ± c (n = even or odd).
for even and odd quotients being alternative
to what is positive and negative in the normal (ii) If (x, y) be the solution of b y = a x + 1, then
cases, leading to the calculation of gua (x) (c x, c y) is the solution of b y = a x + c.
and kepa (c) [evam ameuviamesuam
(iii) If (x , y ) is the solution of b y = a x – c , then
dhanamdhanam  am yaduktam tat / 1 1
(b - x , a - y ) is a solution of b y = a x + c . This
adhanoyor vyastatvamguyaprakepayoh 1 1
was already explained before by Bhāskara I.
kāryam // BSS, xviii. 13];
Likewise, if (x ,y ) is the solution of : b y =
1 1
(iii) Prthudakasvāmi (860 CE) observes that it a x + c , then (b - x , a- y ) is the solution of :
1 1
is not absolute, rather optional, so that the b y = a x - c .
process may be conducted in the same way
by starting with the division of the divisor B. Examples:
corresponding to the smaller remainder by the
divisor corresponding to the greater remainder. 1. To solve : 23 y = 63 x + 1
But in the case of inversion of the process, he
Bhāskara II says that for solution of
continues, the difference of the remainders
23 y = 63 x + 1, the dividend 63 and divisor 23
may be made negative.
are to be mutually divided as in HCF process
Brahmagupta followed the earlier tradition till the remainder reduces to 1, then place the
and his method is no different than the method quotients one below the other with c and 0, as is
of Āryabhata I and Bhāskara I. He clarified the done for mati and final quotient by other authors.
method with a few examples from astronomical Obviously,
and mathematical problems. The most important
contribution of Brahmagupta lies in the fact that he → = 2 + = (2, 1, 2, 1, 1/0)
utilized the knowledge of continued division for = . This gives : 63. 4 – 23. 11= - 1, or 23. 11 =
solution of Vargaprakti of the : N x2± c = y2.
63 x + 1 (no. of quotients = odd); then, x = 4, and
y = 11 gives the solution of 23 y = 63 x + 1;
2.4. Bhāskara II (b.1114 CE)
Bhāskara II, a versatile scholar from the 2. To solve : 63 y = 100 x + 13
school of ujjain in the field of mathematics and
→ = 1 + = (1, 1,
astronomy was trained by astronomer father
Mahevara at Bijjalabia under the patronage of 1, 2, 2, 1, 13/0)= (penultimate convergent);
aka king I. His Bījagaita contains important This gives : 100. 221- 63. 351= - 13 (n = odd) ,
contributions in algebra. or 63. 351 = 100. 221 + 13;x = 221 = 63. 3 + 32
8 INdIAN JourNAL of HIStory of SCIENCE
= 32 (mod 63); y = 351 = 100. 3 + 51 = 51 (mod this implies that Nārāyaa, following
100). Hence x = 32, y = 51 is the least solution of others, has categorically said that the solution lies
63 y = 100 x + 13. in the approximation of √N →
The complete theory of solutions was
3. Vargaprakti expounded by Euler and Lagrange later in 1767
3.1. Definition:The Vargaprakti involves solutions CE.
of indeterminate equations of the type : Nx2±k =
y2, where 3.2 Brahmagupta’s Solutions
N → a non-square integer, known as prakti or Brahmagupta’s solutions in rational
integers of both positive and negative types
guaka;
of the equation Nx2± k = y2, may be explained
x → known as lesser root, kanihapada,
with method of cross-multiplication, known as
hrasvamūla, or ādyamūla,
Bhāvanā or Lemmas.
y → refers to greater root, jyehamūla, anyamūla, Lemma I: Brahmagupta (BSS, xviii, 64-65) first
or antyamūla and formed a set of auxiliary equations described
as follows:
k → refers to number added, kepa, prakepa, or
prakepaka. mūlam dvidhā iavargād guakaguād ia yuta
vihinān ca /
Brahmagupta obtained two sets of
approximate values and applied the process of ādyavadho guakaguah saha antyaghātena ktam
Bhāvanā. Jayadeva and Śrīpati (both of 11th antyam //
century CE) established the process of Cakravāla
vajravadhaikam prathamam prakepah
and led the foundation, while Bhāskara II (c. 1150
kepabadhatulyah/
CE) and Nārāyaa (c. 1350 CE) made further
prakepaśodhakahte mūle prakepake rūpe//
extention and clarification with examples in his
Gaita Kaumudi (Gk). The solution however
English Translation:
is based on the theory of continued fraction as
expounded by Āryabhaa I, Bhāskara I. ‘From the square of an assumed number
multiplied by the guaka, add or subtract a
Nārāyaa following tradition had catego-
desired quantity and obtain the root, and place
rically said which runs thus
them twice. the product of the first [pair of roots]
mūlam grāhyam yasya ca tadrūpakepake multiplied by the guaka increased by the product
of the last [pair of roots] is the [new] greater root
pade tatra /
(antya-mūlam). The sum of the products of the
jyeham hrasvapadena ca samuddhan
cross-multiplication (vajravadhaiam) is the first
mūlamāsannam // [new] root (prathama-mūlam). The [new] kepa
is the product of similar additive or subtractive
“ Obtain the roots (of the Vargaprakti)
quantities. When the kepa is equal (tulya), the
with kepa quantity as unity (i.e., N x2+ 1 = y2)
root [first or last] is to be divided by it to turn the
and the number (N) whose square-root is to be
[new] kepa into unity’.
obtained; then the greater root divided by the
smaller root will determine an approximate value This explains Samāsa (additive), Viślea
of the square-root (√N)’’ (Gk, p.244). (subtractive) and Tulyabhāvanā (equal roots)
SoME fEAturES of tHE SoLutIoNS of KuaKa ANd VarGapraktI 9
discussed under features (A&B).
prakti Kaniha root Jeha root Kepa
Lemma II : Brahmagupta (BSS, xviii. 65) says, N a b k
if x = a, y = b be a solution of : N x2 + k2= y2,
a b k
then x = a/k , y = b/k , is the solution of : N
x2+ 1= y2. then it satisfies, : N(2ab)2 +k2 = (N a2 +
b2)2. By application of Lemma II, it is reduced to:
Lemma III: Brahmagupta (BSS, xviii, 66-69)
N ( )2 + 1 = { }2.The aim was to obtain
prescribed his subsequent rules, which
explains how the solution of the equation the solution of N x2 + 1 = y2, so on.
N x2 + k = y2 is obtained when k.= ±1,±2,±4
C. Example (Brahmagupta):
by applying tulyabhāvanā.
Brahmagupta gave several examples of
Features: Lemma I suggests the following: which one is to solve
(A) Samāsa and Viślea Bhāvanā: 92x2 + 1 = y2 (Brahmasphuasiddhnta
(Dvivedin 1902, BSS, xviii, 75), where x refers
If, (a ,b ,k ) and (a ,b ,k ) satisfy the
1 1 1 2 2 2 to the rāśiśea, y to the ahargaa of the planet
equations of the type: Nx2 ± k = y2 by choice,
Mercury, and N = 92.
then put
(i) For solution of the example, select 92. (1)2
+ 8 = (10)2, then tulya bhāvanā is applied as
prakti Kaniha root Jeha root Kepa
follows:
N a b k
1 1 1
a b k prakti Kaniha root Jeha root Kepa
2 2 2
92 1 10 8
[then it satisfies] N (a b ± a b )2 + k k
1 2 2 1 1 2 1 10 8
= (Na a ± b b )2 i.e. x = Kaiha root = (a b New root
1 2 1 2 1 2 20 192 64
± a b ) , y = Jyeha root= (Na a ± b b ) will
2 1 1 2 1 2
satisfy the equation, N x2+ k k = y2. New Equation:
1 2
It will satisfy both the addition (samāsa- (2) 92( )2 + 1 = ( )2; or, 92( )2 + 1=(24)2
bhāvanā) and the subtraction rule (vilea-
Then again repeating the process of tulya-
bhāvanā). This was discovered by Brahmagupta,
bhvana, we get:
and later rediscovered by Euler in 1764. This also
leads to Tulya Bhāvanā when both the roots are prakti Kaniha root Jeha root Kepa
same.
5/2 24 1
(B) Tulya Bhāvanā (when two roots are equal), 92 5/2 24 1
which is a special case of samsa-bhvana. The
New Roots 120 1151 1
rule runs as follows:
If (a, b, k) and (a, b, k) the two equal roots (3) New roots: . 24+ . 24 =120; 92. + 24.24 =
of Nx2 + k = y2 is taken into consideration by
1151; 1.1 = 1.The roots satisfy, 92 (120)2 + 1 = (1151)2
choice, then put twice the roots, which gives the required solution. When compared
10 INdIAN JourNAL of HIStory of SCIENCE
with the original equation : N x2 + 1 = y2, then Jayadeva developed a new set of auxiliary roots
x = 120, y = 1151, N being 92.; by Cakravāla as follows:
the convergents of √92 → = , prakti Kaniha root Jeha root Kepa
The solution is obtained in 3rd step. Brahmagupta’s N a b k

method of solution of N x2 +1= y2 is, no doubt 1 m m2-N
Na + bm k
interesting but limited, and based on arbitrary (new root) am+b
(m2-N)
choice.
The new root satisfies the equation :
4. SolutIon of VargapraKtI by JayadeVa N (am + b)2 + k(m2 —N) = (Na + bm)2
(1100 ce) and otherS
dividing by k2 we get,
The Cakravāla process, an improved
(b) N { }2 + = { }2
method, was first given by Jayadeva and Śrīpati
in the eleventh century, followed by Bhāskara II In verses 9-11, Jayadeva also hinted at a
(1150 CE), Nārāyaa Pandita (c. 1350 CE) and ready made new kaniha (lesser) root in the form
others.
of a kuaka i.e., , a new jyeha (greater)
4.1. Solutionof :Nx2 + 1 = y2 by the Cakravāla root = , and a new kepa =/ /. He
(Cyclic) Process
said that they should be integers and that the value
The Sundarī, udayadivākara’s commentary of m should be so selected that the new kepa
on the Laghubhāskarīya of Bhāskara1 (Vargaprakii,
should be an integer as small as possible.
verses 8-15) quotes from Jayadeva’s work1. This
As regards new kepa,/ , Ācārya
was brought to our notice by Shukla (1954) :
Jayadeva said, tāvat kteh praktyā hine prakepakena
Jayadeva assumed in verse 8 one set of
sambhakte svalpatarā avāpti syāt ityakalitā aparah
integer values (a, b, k) for lesser (kaniha) root,
kepa (verse 9) i.e., tāvatkteh (m2) praktyā hine
greater (jyeha) root and kepa number satisfying
diminished by (N) and prakepakena sambhakte
Nx2 + k = y2 , then found the other set (1, m, k)
divided by the interpolator (k), should be such that
satisfying the identity equation: N. 12 + (m2- N)
it yields the least value (svalpatarā avapti syāt).
= m2 where kepa quantity, k = (m2- N).
As regards new kaniha (lesser) root
The process of Bhāvanā is then applied, by
Jayadeva to find an arbitrary set, as follows: , he said, prakipta-prakepa-kuakāre
(a) taking kaihamūlahate sajyehapade prakep(ak)
Na2 + k =b2 and ea labdham kaihapadam / (verse 10 ).i.e.,
kaihapadam lesser root is obtained (labdham)
N.12 + (m2- N)= m2 (an identity),
1 hrasvajyehakepān pratirāśya kepabhaktayoh kepāt / kuakāre ca kte kiyadguam kepakam kiptvā // (8)
tāvatkteh praktyā hīne prakepakena sambhakte / svalpatarāvāptih syād ityakālito 'parah kepah //(9)
prakiptaprakepakakuakāre kaihamūlahate / sajyehapade prakep(ak)ea labdham kaihapadam //(10)
k iptakepakakuaguitāt tasmāt kaihamūlahatam / pāścātyam prakepam viśodhya. śeam mahānmūlam 1/ (11)
kuryāt kuakāram punar anayoh kepabhaktayoh padayoh/ tat .sa iahatakepe sadsague 'sminprakrtihīne // (12)
prakepah kepāpte prakiptakepakāc ca guakārāt / alpaghnāt sajyehāt kepāvāptatn kaihapadam // (13)
etas kiptakepakakuakaghātādanatarakepam / hitvā 'Ipahatam śeam jyeham tebhyaś ca guakādi (14)
kuryād tāvad yāvat saāmekadvicaturnām patati / iti cakravāla karae ‘vasaraprāptāniyojyāi (15).